Noether's Theorem on Superphysics

Invariant variational problems

Mon, 01 Jan 0001 00:00:00 +0000

We shall deal with variational problems that admit a continuous group (in the Lie sense).

Two Theorems

Mon, 01 Jan 0001 00:00:00 +0000

One now deals with the two theorems in what follows:

I. If the integral I is invariant under a Gρ then there will be ρ linearly independent couplings of the Lagrangian expressions with divergences.

Divergence relations and dependencies.

Mon, 01 Jan 0001 00:00:00 +0000

Let G be a finite or infinite group.

We may then always arrange that the identity transformation corresponds to the value zero for the parameter s (the arbitrary functions p(x), resp.) 4). The most general transformation then takes the form:

Linearly-independent Couplings

Mon, 01 Jan 0001 00:00:00 +0000

One thus has ρ linearly-independent couplings of the Lagrangian expressions with divergences; the linear independence follows from the fact that, from (9), it would follow that δ u = 0, ∆u = 0, ∆x = 0, so there would be a dependency between the infinitesimal transformations.

Converse in the case of the finite group.

Mon, 01 Jan 0001 00:00:00 +0000

In order to show the converse, one must essentially follow through the foregoing argument in the opposite sequence. The validity of (12) follows from the validity of (13) upon multiplication by ε and addition, and by means of the identity (3), this implies a relation: δ f + Div(A – B) = 0. If one then sets: ∆x = 1/f ⋅ (A – B) then one arrives at (11) as a result of this.

Linearly-independent Couplings

Mon, 01 Jan 0001 00:00:00 +0000

§ 4. Converse in the case of infinite groups.

First, let us show that the assumption of the linearity of ∆x and ∆u presents no restriction, which one deduces here without the converse from the fact that G∞ρ formally depends upon ρ and only ρ arbitrary functions. Namely, it shows that in the nonlinear case the number of arbitrary functions would increase under the composition of transformations in which the terms of lowest order would add together. In fact, let, say:

Invariance of the individual components of the relations.

Mon, 01 Jan 0001 00:00:00 +0000

If one specializes the group G to the simplest case that is ordinarily considered by specifying that one allows no derivatives of the u in the transformations and that the transformed independent variables depend upon only the x, but not the u then one can deduce the invariance of the individual components in the formulas.